Miller CJ et al. (2018), Emergent mechanics of actomyosin drive punctuat...

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PLoS Comput Biol 2018 Sep 01;149:e1006344. doi: 10.1371/journal.pcbi.1006344.

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Emergent mechanics of actomyosin drive punctuated contractions and shape network morphology in the cell cortex.

Miller CJ , Harris D , Weaver R , Ermentrout GB .

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Filamentous actin (F-actin) and non-muscle myosin II motors drive cell motility and cell shape changes that guide large scale tissue movements during embryonic morphogenesis. To gain a better understanding of the role of actomyosin in vivo, we have developed a two-dimensional (2D) computational model to study emergent phenomena of dynamic unbranched actomyosin arrays in the cell cortex. These phenomena include actomyosin punctuated contractions, or "actin asters" that form within quiescent F-actin networks. Punctuated contractions involve both formation of high intensity aster-like structures and disassembly of those same structures. Our 2D model allows us to explore the kinematics of filament polarity sorting, segregation of motors, and morphology of F-actin arrays that emerge as the model structure and biophysical properties are varied. Our model demonstrates the complex, emergent feedback between filament reorganization and motor transport that generate as well as disassemble actin asters. Since intracellular actomyosin dynamics are thought to be controlled by localization of scaffold proteins that bind F-actin or their myosin motors we also apply our 2D model to recapitulate in vitro studies that have revealed complex patterns of actomyosin that assemble from patterning filaments and motor complexes with microcontact printing. Although we use a minimal representation of filament, motor, and cross-linker biophysics, our model establishes a framework for investigating the role of other actin binding proteins, how they might alter actomyosin dynamics, and makes predictions that can be tested experimentally within live cells as well as within in vitro models.

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	Fig 1. Kinetics and diverse morphology of actomyosin contractions in Xenopus laevis embryos.(A) Punctuated F-actin contractions, or "actin-asters" are observed the apical cortex of Xenopus neural epithelial cells during gastrulation. F-actin is seen across the apical surface and cell-cell junctions but is transiently enriched during a contraction (circle). A series of frames from a time-lapse sequence (arrow) reveals rapid accumulation and dissipation of F-actin in the apical cortex. A kymograph of the contraction (across blue line) shows the changing intensity of F-actin and quantified as changes in normalized intensity (Icontraction/Icell) over length of the time lapse. Normalized intensity is based on identifying a region of interest (ROI) and tracking the intensity with that ROI over time. (B) Frames from a time-lapse series where F-actin (mChe-life-act) and mini-thick filaments of active myosin II (MyoII; mNeon-sf9, [22]) can be tracked in the basal cell cortex. Temporal analysis reveals that myosin co-localizes with F-actin asters but lags both accumulation and dissociation profiles of F-actin. (C) Similar spatial and temporal patterns of F-actin are evident on the basal surface of a Xenopus animal cap explant cultured on fibronectin coated glass, (D) the apical surface of an animal cap explant cultured against a clean glass surface, (E) the apical surface of the blastopore lip (confocal image provided by Joseph Shawky and Rafey Feroze, personal communication, March 2014), and (F) across a broad field of cells of the neural epithelium (confocal image provided by Deepthi Vijayraghavan, personal communication, June 2017).
	Fig 2. Dynamic model of filaments and motors in two dimensions.(Aâ€”E) Schematic depicting the biophysical interactions and dynamics of F-actin and myosin II motors. A) G-actin monomer extends the plus- or barbed-end of the actin filament through polymerization and disassembles the filament at the minus end. Our simulation does not explicitly implement polymerization or depolymerization dynamics but rather captures the impact of those processes through a single turn-over rate, i.e. one filament completely depolymerizes and a new one polymerizes at a new random location. (B) Individual non-muscle myosin II motors bundle together into a multi-headed mini thick filament and when activated bind to F-actin, or when inactivated detach from F-actin. Our simulation does not explicitly implement motor assembly, activation, or inactivation. Instead, our model motors represent non-muscle myosin II already bundled into activated myosin filaments. (C) Once bound, myosin filaments 'walk' to the plus-end of the F-actin. (D) Vector diagram of force generated by a single motor attached to a filament. The filament bound by the motor moves in 2D as vector forces at the attachment point are decomposed into parallel and perpendicular forces which are opposed by viscous drag. The perpendicular force (blue) leads to a perpendicular translation and torque applied to the ends drive filament rotation; forces applied parallel to the filament drive translational movement. (E) Schematic showing filament movements after a searching motor binds a pair of filaments: after the motor binds a pair of filaments each head domain moves at a fixed velocity toward the bound filament's plus-end. After one time step, the motor exerts a force couple on the filament pair, rotating and pulling the filaments together. Repeated applications of such directed forces result in polarity sorting of filaments that may have initially been anti-parallel. (F) A sparse network of 50 filaments (white) and 250 motors (not shown) illustrates filament movements over 1,000 time steps or 10 seconds. (G) Filament divergence, a measure of aster assembly, reveals a single aster at point of transition from low to high divergence. (H) The mean motor exerted force (blue; pN) over 1,000 time steps or 10 seconds. (I) The time-evolution of mean motor exerted force for 100 simulations, each initiated with randomly positioned filaments and motors.
	Fig 3. Actomyosin arrays remodel in simulated dense networks.(A) Sequence of images from a simulated time lapse of 1,000 actin filaments (red) and 5,000 myosin motors (green). (B) The normalized actin mean intensity (purple) within a region in A (circle at 0 s). The mean intensity is based on identifying the location of an aster within a region of interest (ROI; in this case a circle), and tracking the intensity in that ROI over time. (C) Kymograph at top show formation of an aster over 10 seconds (line in A at 10 s). Kymograph below shows a case where the filament turnover rate is high (p2, 5/s) and no asters are formed. (D) The mean motor exerted force quickly peaks and subsides in cases where asters form (pN, red) but remain constants for cases where no asters form (blue). We have compared the mean motor force for 10 simulations with the standard parameter set and observe that the profiles are similar (S2 Fig) (E) The 'connectedness' of filament plus-ends shown over the length of simulation. (F) The divergence of filaments at the conclusion of a simulation generating an aster; high divergence (yellow region in the center) indicates where filament polarity reverses. (G) Our coarse-grained image analysis technique shows highlighted hexagonal areas within the larger domain at 10 s (S4 Video). We quantified the number of the highlighted areas over time (orange). Additional calculations, including a comparison between the image analysis of filament intensity mappings versus divergence mappings are in the supplementary material (S3 Fig).
	Fig 4. Diverse actomyosin arrays can be formed by discrete changes in filament dynamics.Steady state F-actin morphology after 10 s as F-actin biophysics is altered. Arrays from different simulations in the center column (yellow box) are formed by a standard set of parameters. Each row depicts arrays formed when a single parameter regulating F-actin is changed. Arrays formed from simulations are shown in each column and are ordered from lower (left) to higher parameter values (right). Only the parameter described by row and column were varied, the other parameter values were held at the standard value. See S1 Table for the details of parameters used here. We have included an asterisk (*) to note cases where an aster has not formed. To determine if an aster could form over longer times, we ran these simulations for a more extended duration and found only minor changes in the asters (S5 Fig).
	Fig 5. Diverse actomyosin arrays can be formed by discrete changes in motors dynamics.Steady state F-actin morphology after 10 time steps as myosin II biophysics is altered. The yellow box indicates the same set of standard parameters but random initial configuration. Each row depicts morphology generated by changes in a single parameter controlling myosin II motors, ordered from lower value on the left to higher value on the right. Each simulation starts with randomly oriented and positioned filaments and motors. Red numbers represent values outside of the physiological determined range, green numbers are values within physiological range, and white represent numbers whose ranges are best guess values. Rates of motor attachment (p1) and detachment (p0) are estimated. See S1 Table for additional parameters. We have included an asterisk (*) to note cases where an aster has not formed. To determine if an aster could form if given enough time, we ran these simulations for longer time and present results in S5 Fig.
	Fig 6. Changing conditions that destabilize or rescue aster formation.(A) To determine whether asters could be destabilized we ran simulations where conditions that produced stable asters were changed to conditions that did not. Conditions for simulations forming quasi-stable asters after 1,000 time steps (center) were changed and the network was followed for 1,000 additional time steps (left or right column). Stable asters are resistant to disruption under most changes except when rates of filament turn-over are increased or when motor attachment rates are reduced. (B) To determine whether asters could be rescued from non-steady states we ran simulations starting with conditions that did not produce stable asters or produced multiple asters before switching to conditions that only formed single asters from initially disorganized networks. Parameters producing the starting morphology after 1,000 time steps are listed in the center columns while the altered parameters are listed in the right (lower) or left (higher) columns. Single aster morphologies could be rescued by changes in motor properties of stiffness or stretch length but could not be rescued by changes in filament length. In all cases the morphology of filament arrays remodeled after 2,000 time steps are shown either under the â€œLower Parameterâ€ or â€œHigher Parameterâ€ columns.
	Fig 7. Simulating dynamics of actomyosin networks reconstituted on patterned substrates.(A) Time-evolution of asters formed near a fixed âbarâ of actin. The simulation begins with 50% of filaments located entirely inside the bar and 50% of filaments with their minus ends on the bar and plus or barbed ends outside the bar. The density of filaments and motors are the same as previous simulations (1,000 filaments and 5,000 motors). Motors are initially distributed over the entire domain (bar and non-bar). Over time, the motors bind to the filaments and traffic towards filament plus-ends, accumulating within the bar. Abundant motors then contract the F-actin network into an aster. We note: to recreate conditions in the reconstitution studies, filaments in this simulation are stabilized against depolymerization and do not turnover. (B) Time-evolution of asters near tethered motors after a fraction of motors are âtetheredâ to the bottom eighth of the hexagon. This means that one leg of each motor located in the bottom eighth of the hexagon is fixed in place, leaving the remaining leg free to bind filaments. The freely diffusing motor population interacts with filaments throughout the hexagonal domain. Over time, the fixed-position motors pull asters toward the domain where they are tethered.
	Fig 8. Generic actin cross-linkers slow but do not prevent aster formation.Time-evolution of filaments (red), actin cross-linkers (yellow; 1,250) and motors (3,750, not shown) show cross linkers do not inhibit formation of asters. Cross-linkers bind at the same rate as motors (p0) to filament pairs that are oriented within 22.5Â° (Ï/8) of each other and no more than 40 nm apart. The cross linkers bind to the shortest distance between candidate filament pairs, and remain fixed. The merged images in the bottom row show the formation of an aster over the time course. Biophysical parameters guiding filaments and motors in these simulations are the standard set (see S1 Table for details).
	Fig 9. Generic cross linkers do not stabilize aligned F-actin arrays.To test whether cross-linkers (yellow) can stabilize previously aligned filaments (red) and inhibit aster formation we prepared an aligned array of F-actin for the start of a simulation. Filament plus-ends are initially distributed randomly throughout the hexagon with their orientation between 0 and 22.5Â°. Cross-linkers bind at the same rate as motors (p0) to filament pairs that are within Ï/8 degrees of each other and no more than 40 nm apart. The simulation was run with 1,250 cross-linkers and 3,750 myosin motors (not shown). The filament asters form, albeit with distinct intermediate morphologies despite the prior alignment of filaments. Biophysical parameters guiding filaments and motors in these simulations are the standard set (see S1 Table for details).
	Fig 10. Qualitative maps of phase transitions in actomyosin arrays.(a) Proposed model for how disordered actomyosin arrays transition to quasi-static actin asters. Viscous dissipation leaves the system as motor ATP hydrolysis generates reorganization of the actin network. For a quasi-static actin aster to revert back to a disordered state, we believe there is a dissociation of the network resulting from a loss of elastic energy. (b) Phase transitions in parametric space are determined by physical principles of work energy of a dissipative system of filaments, motors, and their viscous environment. For instance, high rates of force production lead to greater losses to viscous dissipation and less contribution to elastic remodeling of the filament array. Aster morphologies (region II) can emerge from disordered arrays (region I) via several mechanisms including reduced filament turnover, reduction of viscous losses, or increased motor-work. In special cases where filaments are shortened (dashed domain, region III) multiple asters can form. Multiple asters are also sensitive to work energy and can merge into single asters (transition to region II) or disperse into disordered arrays (transition to region I) under similar conditions that mediate aster/no aster transitions. The asterisk indicates the location of our âstandardâ parameter set leading to a single robust aster.

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